Comparison of features on Full and Cropped Scans

TL;DR

To conduct our analysis, we did a series of tests to compare the cropped version of a scan against the full version. We constructed scatter and box plots as well as ROC curves for visual analysis We trained a Generalized Linear Model (glm) to test the p scores, or importance, of each variable in the model, and constructed kernel density graphs to compare the scans. Every cropped scan and its full scan counterpart were found to have more than 90% correlation with each other, requiring us to use only one of them for each feature to avoid co-linearity issues.

Extract NA, a measure of the total missing value count in the scan over the total value count. This feature performed better when it is cropped with no reservations. The ROC curve and p score in the glm were both better for the cropped image. The kernel density graph was also more distinct for the cropped image.

Assess Bottomempty, a measure of missing value count in the bottom 20% of the scan compared to the total value count in the same area. This feature performed better when it is cropped with no reservations. The ROC curve and p score in the glm were both better for the cropped image. The kernel density graph was also more distinct for the cropped image.

Assess Col NA, the proportion of columns in the scans matrix which have more than 20% missing values. This feature performed only marginally better when cropped. The ROC area difference is only 0.008, and the p scores for both features were found to be extremely significant. The kernal density graphs are very similar with the cropped graph being slightly more distinct.

Assess Median NA Proportion, calculates the mean number of NA’s in each column, and then finds the median out of all those values. This feature performed better when left as the full scan. The ROC area for the cropped was better by only 0.001, the p values for both features were found to be extremely significant, but the full scan was lower. The kernel density graph was more distinct for the full images.

Correlation pvalue_Full pvalue_cropped auc_Full auc_Cropped
Extract NA 0.908 0.229 <2e-16 0.871 0.902
Assess Bottomempty 0.905 5.86e-10 <2e-16 0.783 0.859
Assess Col NA 0.919 1.62e-06 1.17e-14 0.888 0.896
Assess Median NA Proportion 0.908 <2e-16 6.77e-10 0.907 0.863

All of the features had significant outliers, which we have called “Followup Scans”, for their individual predictive power. We investigated the type I errors, false positives of a bad scan being predicted as good, and found them all to be “Tiny Problems” scans which could reasonably be re-classified as “problematic” or worse.

Background and Introduction

This analysis is to compare the difference between the cropped versus non-cropped (full) version of a scan for quality identification. Cropped images have the potential for decreasing noise around the signal. The level of cropping we are considering is 5% from the left and right sides, and 10% off of the top of the image. In particular, we want to preserve the bottom of the image and the center as that is where most of the signal is. Below are examples of a full, full with marked edges, and a cropped image.

Image of a scan without cropping Image of a scan with the edges colored Image of a scan with cropping

Features

Each of the features shares a similar set up, so the following assumptions and definitions will remain independently true for each of the features defined below.

Let \(A=\{NA\}\) be the set of undefined values. For simplicity of notation we will assume that the space of real values \({\rm I\!R}\) contains \(A\): \({\rm I\!R}:= {\rm I\!R} \cup A\).

Let \(X \in R^{m,n}\) be a real-valued surface matrix of dimensions m x n where m and n are strictly positive integers \(X = (x_{ij})_{1 \leq i \leq m, 1 \leq j \leq n}\).

Extract NA

The function extract_na calculates the percentage of missing values in the scan (part) under observation, e.g. for scan surface matrix \(X \in {\rm I\!R}^{m, n}\) the percentage of missing values is defined as:

The proportion of missing values in X is then defined as: \[ \frac{1}{m*n} \sum^m_{i=1} \sum^n_{j=1} \theta_A(x_{ij}) \\ \text{Where } \theta_A(x) = \left\{\begin{aligned} &1 &&: \text{if }x \in A\\ &0 &&: \text{otherwise}\\ \end{aligned} \right. \] Assess Bottomempty

The feature assess_bottomempty calculates the percentage of missing values in the bottom 20% of the scan.

Let \(R \subseteq {\rm I\!R}\) be a set of size m, where each element is the sum of the NA’s for the given row, defined as: \[ \forall i \in R: R_i = \sum^n_{j=1} \theta_A(x_{ij}) \\ \text{Where } \theta_A(x) = \left\{\begin{aligned} &1 &&: \text{if }x \in A\\ &0 &&: \text{otherwise}\\ \end{aligned} \right. \]

Let \(B \subset R\) be a set, which is the set of all values in \(R_i\), given that \(i \geq m*0.8\). Therefore, the proportion of missing values in \(X\)’s bottom 20% can be given by: \[ \frac{1}{m*n*0.2}\sum_{i=1}^{m*0.2}(R_i)*100 \] Assess Col NA

The function assess_col_na calculates the percentage of missing values

For every column in the matrix of a scan, we find the proportion of scans in that column which are NA. Then we count how many of the columns whose proportion is greater than 20%, the pre-determined threshold of acceptable NA’s. Then we divide by the number of columns * 0.2 to get our final threshold adjusted number.

Let \(R \subseteq {\rm I\!R}\) be a set of size n, where each element is the sum of the NA’s for the given column, defined as: \[ \forall i \in R: R_i = \sum^m_{j=1} \theta_A(x_{ij}) \\ \text{Where } \theta_A(x) = \left\{\begin{aligned} &1 &&: \text{if }x \in A\\ &0 &&: \text{otherwise}\\ \end{aligned} \right. \]

We define \(P\) as the proportion of all NAs per column for every row, as defined here: \[ \forall i \in R: P_i = \frac{R_i}{n} * 100 \]

We now find the proportion of threshold adjusted columns in the matrix \[ \frac{\sum_{i=1}^n(P_i*\beta_B(P_i))}{n*0.2} \\ \text{Where } \beta_B(x) = \left\{\begin{aligned} &1 &&: \text{if }x > 20\\ &0 &&: \text{otherwise}\\ \end{aligned} \right. \]

Assess Median NA proportion

The function assess_median_na_proportion calculates the mean number of NA’s in each column, and then finds the median out of all those values.

Let \(R \subseteq {\rm I\!R}\) be a set of size n, where each element is the mean of the NA’s for the given column, defined as: \[ \forall i \in R: R_i = \frac{\sum^m_{j=1} \theta_A(x_{ij})}{m} \\ \text{Where } \theta_A(x) = \left\{\begin{aligned} &1 &&: \text{if }x \in A\\ &0 &&: \text{otherwise}\\ \end{aligned} \right. \]

We then sort, and select the median of \(R\)

Appendix

Extract NA

Which of the features is better for discriminating between good and bad scans?

## [1] "Extract NA. Correlation:  0.908 Full AUC: 0.871 Cropped AUC:  0.902"
Summary Info of Extract NA
min firstQ med mean thirdQ max
Standard 3.666645 12.79727 15.082281 15.915471 18.30296 48.45372
Cropped 1.389464 6.38338 8.060005 8.916349 10.57698 40.80471

Should we use features from just one type of scan or both?

## 
## Call:
## glm(formula = GoodScan ~ extract_na + extract_na_cropped, family = binomial(), 
##     data = full_data)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -3.1882  -0.2394   0.3044   0.5298   4.0795  
## 
## Coefficients:
##                    Estimate Std. Error z value Pr(>|z|)    
## (Intercept)         7.16247    0.37858  18.919   <2e-16 ***
## extract_na         -0.04349    0.03613  -1.204    0.229    
## extract_na_cropped -0.58303    0.05020 -11.615   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2230.6  on 1849  degrees of freedom
## Residual deviance: 1299.2  on 1847  degrees of freedom
## AIC: 1305.2
## 
## Number of Fisher Scoring iterations: 6

Conclusion for Extract NA

The values for feature extract_NA are highly correlated between the cropped and the full scan.

Using good and scans with only tiny problems as overall ‘good’ scans, the feature applied to cropped scans has an increased accuracy compared to the feature values from the full scan.

We might want to follow up on the orange colored scans:

full_data$LAPD_id[full_data$followup]
## [1] "FAU263-BA-L4" "FAU263-BC-L1" "FAU263-BC-L3" "FAU287-BC-L5" "FAU154-BD-L2"
## [6] "FAU277-BA-L4" "FAU286-BA-L5"
followupScans <- rbind(followupScans, full_data[full_data$followup == TRUE,])
# All followups for extract_na are mislabelled scans. They are all labelled as tiny problems but should be problematic or worse.

Assess Bottomempty

Which of the features is better for discriminating between good and bad scans?

## [1] "Assess Bottomempty. Correlation:  0.905 Full AUC: 0.783 Cropped AUC:  0.859"
Summary Info of Assess Bottomempty
min firstQ med mean thirdQ max
Standard 8.516493 22.66947 27.54019 29.90259 34.63373 95.39375
Cropped 3.564558 10.78581 13.54205 15.28335 17.71191 84.22476

Should we use features from just one type of scan or both?

## 
## Call:
## glm(formula = GoodScan ~ assess_bottomempty + assess_bottomempty_cropped, 
##     family = binomial(), data = full_data)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.6794  -0.2813   0.4084   0.6028   4.0680  
## 
## Coefficients:
##                            Estimate Std. Error z value Pr(>|z|)    
## (Intercept)                 4.57647    0.25156  18.193  < 2e-16 ***
## assess_bottomempty          0.09046    0.01460   6.194 5.86e-10 ***
## assess_bottomempty_cropped -0.40779    0.02738 -14.895  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2230.6  on 1849  degrees of freedom
## Residual deviance: 1503.7  on 1847  degrees of freedom
## AIC: 1509.7
## 
## Number of Fisher Scoring iterations: 5

Conclusion for Assess Bottomempty

The values for feature assess_bottomempty are highly correlated between the cropped and the full scan.

Using good and scans with only tiny problems as overall ‘good’ scans, the feature applied to cropped scans has an increased accuracy compared to the feature values from the full scan.

We might want to follow up on the orange colored scans:

full_data$LAPD_id[full_data$followup]
## [1] "FAU263-BA-L4" "FAU287-BC-L5" "FAU254-BD-L4" "FAU275-BC-L5" "FAU275-BD-L3"
## [6] "FAU277-BA-L4" "FAU286-BA-L5"
followupScans <- rbind(followupScans, full_data[full_data$followup == TRUE,])

Assess Col NA

Which of the features is better for discriminating between good and bad scans?

## [1] "Assess Col NA Correlation:  0.919 Full AUC: 0.888 Cropped AUC:  0.896"
Summary Info of Assess Col NA
min firstQ med mean thirdQ max
Standard 0.2573318 0.9244176 1.0755102 1.1654392 1.292844 4.368436
Cropped 0.0920783 0.4700281 0.6126759 0.6851693 0.807577 3.356120

Should we use features from just one type of scan or both?

## 
## Call:
## glm(formula = GoodScan ~ assess_col_na + assess_col_na_cropped, 
##     family = binomial(), data = full_data)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -3.2254  -0.1897   0.3176   0.5365   3.6049  
## 
## Coefficients:
##                       Estimate Std. Error z value Pr(>|z|)    
## (Intercept)             7.3581     0.3785  19.440  < 2e-16 ***
## assess_col_na          -2.4724     0.5155  -4.796 1.62e-06 ***
## assess_col_na_cropped  -4.8233     0.6249  -7.719 1.17e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2230.6  on 1849  degrees of freedom
## Residual deviance: 1308.8  on 1847  degrees of freedom
## AIC: 1314.8
## 
## Number of Fisher Scoring iterations: 6

Conclusion for Assess Col NA

The values for feature assess_col_na are highly correlated between the cropped and the full scan.

Using good and scans with only tiny problems as overall ‘good’ scans, the feature applied to cropped scans has an increased accuracy compared to the feature values from the full scan.

We might want to follow up on the orange colored scans:

full_data$LAPD_id[full_data$followup]
## [1] "FAU263-BA-L4" "FAU263-BB-L3" "FAU263-BC-L1" "FAU263-BC-L3" "FAU154-BD-L2"
## [6] "FAU286-BA-L5"
followupScans <- rbind(followupScans, full_data[full_data$followup == TRUE,])

Assess Median NA Proportion

Which of the features is better for discriminating between good and bad scans?

## [1] "Assess Median NA Proportion. Correlation:  0.908 Full AUC: 0.907 Cropped AUC:  0.863"
Summary Info of Assess median NA proportion
min firstQ med mean thirdQ max
Standard 0.0000000 0.0033099 0.0108771 0.0207348 0.0271577 0.2552491
Cropped 0.0011692 0.0518201 0.0686747 0.0751096 0.0918769 0.2994505

Should we use features from just one type of scan or both?

## 
## Call:
## glm(formula = GoodScan ~ assess_median_na_proportion + assess_median_na_proportion_cropped, 
##     family = binomial(), data = full_data)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.8226  -0.1079   0.3143   0.4865   5.2612  
## 
## Coefficients:
##                                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)                            4.384      0.257  17.060  < 2e-16 ***
## assess_median_na_proportion          -83.035      5.925 -14.014  < 2e-16 ***
## assess_median_na_proportion_cropped  -21.546      3.491  -6.171 6.77e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 2230.6  on 1849  degrees of freedom
## Residual deviance: 1219.0  on 1847  degrees of freedom
## AIC: 1225
## 
## Number of Fisher Scoring iterations: 6

Conclusion for Assess Median NA Proportion

The values for feature extract_NA are highly correlated between the cropped and the full scan.

Using good and scans with only tiny problems as overall ‘good’ scans, the feature applied to full scans has an increased accuracy compared to the feature values from the cropped scan.

We might want to follow up on the orange colored scans:

full_data$LAPD_id[full_data$followup]
## [1] "FAU263-BC-L3" "FAU154-BD-L2" "FAU204-BC-L4"
followupScans <- rbind(followupScans, full_data[full_data$followup == TRUE,])

Followup Scans

followupUnique <- followupScans[duplicated(followupScans) == FALSE,]
followupScans %>% group_by(followupScans$LAPD_id) %>% summarize(
  count = n()
)
## # A tibble: 12 x 2
##    `followupScans$LAPD_id` count
##    <chr>                   <int>
##  1 FAU154-BD-L2                3
##  2 FAU204-BC-L4                1
##  3 FAU254-BD-L4                1
##  4 FAU263-BA-L4                3
##  5 FAU263-BB-L3                1
##  6 FAU263-BC-L1                2
##  7 FAU263-BC-L3                3
##  8 FAU275-BC-L5                1
##  9 FAU275-BD-L3                1
## 10 FAU277-BA-L4                2
## 11 FAU286-BA-L5                3
## 12 FAU287-BC-L5                2
# 3 hits: FAU154-BD-L2, FAU263-BA-L4, FAU263-BC-L3, FAU286-BA-L5
# 2 hits: FAU263-BC-L1, FAU277-BA-L4, FAU287-BC-L5
# 1 hits: FAU204-BC-L4, FAU254-BD-L4, FAU263-BB-L3, FAU275-BC-L5, FAU275-BD-L3
FAU263_BA_L4 <- x3p_read("../data/followup_scans/LAPD - 263 - Bullet A - Land 4 - Sneox2 - 20x - auto light left image + 20% - threshold 2 - resolution 4 - Connor Hergenreter.x3p")
FAU263_BC_L1 <- x3p_read("../data/followup_scans/LAPD - 263 - Bullet C - Land 1 - Sneox2 - 20x - auto light left image + 20% - threshold 2 - resolution 4 - Connor Hergenreter.x3p")
FAU263_BC_L3 <- x3p_read("../data/followup_scans/LAPD - 263 - Bullet C - Land 3 - Sneox2 - 20x - auto light left image + 20% - threshold 2 - resolution 4 - Connor Hergenreter.x3p")
FAU277_BA_L4 <- x3p_read("../data/followup_scans/LAPD - 277 - Bullet A - Land 4 - Sneox2 - 20x - auto light left image + 20% - threshold 2 - resolution 4 - Connor Hergenreter.x3p")
x3p_image(FAU263_BA_L4, file="./Comparative-Analysis_files/figure-html/FAU263_BA_L4.png")
x3p_image(FAU263_BC_L1, file="./Comparative-Analysis_files/figure-html/FAU263-BC-L1.png")
x3p_image(FAU263_BC_L3, file="./Comparative-Analysis_files/figure-html/FAU263_BC_L3.png")
x3p_image(FAU277_BA_L4, file="./Comparative-Analysis_files/figure-html/FAU277-BA-L4.png")
LAPD.ID Hit.Count Current.Quality Current.Problem Recommended.Quality
FAU263-BA-L4 3 Tiny Problems Feathering NA
FAU263-BC-L3 3 Tiny Problems Feathering NA
FAU154-BD-L2 3 Tiny Problems Holes NA
FAU286-BA-L5 3 Tiny Problems Holes NA
FAU263-BC-L1 2 Tiny Problems Feathering NA
FAU287-BC-L5 2 Tiny Problems Feathering NA
FAU277-BA-L4 2 Tiny Problems Holes NA
FAU254-BD-L4 1 Tiny Problems Holes NA
FAU275-BC-L5 1 Tiny Problems Holes NA
FAU275-BD-L3 1 Tiny Problems Holes NA
FAU263-BB-L3 1 Tiny Problems Feathering NA
FAU204-BC-L4 1 Tiny Problems Holes NA

FAU154_BD_L2 Image

FAU154_BD_L2 (3 hits, Tiny Problems, Holes): Significant feathering across image.

FAU204_BC_L4 Image

FAU204_BC_L4 (1 hits, Tiny Problems, Holes): Feathering on each end of image, rotational issues on left edge, holes in the center

FAU254_BD_L4 Image

FAU254_BD_L4 (1 hits, Tiny Problems, Holes): Significant missing values on the bottom, holes across the center, missing section on right side.

FAU263_BA_L4 Image

FAU263_BA_L4 (3 hits, Tiny Problems, Feathering): Significant feathering on right side image, missing most of the left, and many missing values on bottom

FAU263_BC_L1 Image

FAU263_BC_L1 (2 hits, Tiny Problems, Feathering): Significant feathering on right hand side, left side is missing most of the values, then feathering, then holes as it moves towards the middle. Bottom is also speckled with missing values.

FAU263_BC_L3 Image

FAU263_BC_L3 (3 hits, Tiny Problems, Feathering): Contains significant feathering, holes, disproportionate edges and missing values at the bottom.

FAU275_BC_L5 Image

FAU275_BC_L5 (1 hits, Tiny Problems, Holes):

FAU275_BD_L3 Image

FAU275_BD_L3 (1 hits, Tiny Problems, Holes):

FAU277_BA_L4 Image

FAU277_BA_L4 (2 hits, Tiny Problems, Holes): A few holes, significant missing values on the left, right, and bottom.

FAU286_BA_L5 Image

FAU286_BA_L5 (3 hits, Tiny Problems, Holes):